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1.
Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional
derivative of a continuous function. This space of distributions is denoted
Ac(\mathbb T){\mathcal{A}}_{c}(\mathbb{T}) and is a Banach space under the Alexiewicz norm,
|| f|| \mathbbT=sup |I| £ 2p|ò I f|\|f\|_{\mathbb{T}}=\sup_{|I|\leq2\pi}|\int_{I} f|, the supremum being taken over intervals of length not exceeding 2 π. It contains the periodic functions integrable in the sense of Lebesgue and Henstock–Kurzweil. Many of the properties of
L
1 Fourier series continue to hold for this larger space, with the L
1 norm replaced by the Alexiewicz norm. The Riemann–Lebesgue lemma takes the form [^( f)]( n)= o( n)\hat{f}(n)=o(n) as | n|→∞. The convolution is defined for
f ? Ac(\mathbb T)f\in{\mathcal{A}}_{c}(\mathbb{T}) and g a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative.
There is the estimate
|| f* g|| ¥ £ || f|| \mathbbT || g|| BV\|f\ast g\|_{\infty}\leq\|f\|_{\mathbb{T}} \|g\|_{\mathcal{BV}}. For
g ? L1(\mathbb T)g\in L^{1}(\mathbb{T}),
|| f* g|| \mathbbT £ || f|| \mathbb T || g|| 1\|f\ast g\|_{\mathbb{T}}\leq\|f\|_{\mathbb {T}} \|g\|_{1}. As well, [^( f* g)]( n)=[^( f)]( n) [^( g)]( n)\widehat{f\ast g}(n)=\hat{f}(n) \hat{g}(n). There are versions of the Salem–Zygmund–Rudin–Cohen factorization theorem, Fejér’s lemma and the Parseval equality. The
trigonometric polynomials are dense in
Ac(\mathbb T){\mathcal{A}}_{c}(\mathbb{T}). The convolution of f with a sequence of summability kernels converges to f in the Alexiewicz norm. Let D
n
be the Dirichlet kernel and let
f ? L1(\mathbb T)f\in L^{1}(\mathbb{T}). Then
|| Dn* f- f|| \mathbbT?0\|D_{n}\ast f-f\|_{\mathbb{T}}\to0 as n→∞. Fourier coefficients of functions of bounded variation are characterized. The Appendix contains a type of Fubini theorem. 相似文献
2.
Given k ∈ L1 (0,1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator Vk defined on Lp (0,1) by
and its iterates
We construct some much simpler sequences which, as n → ∞, are asymptotically equal in the operator norm to Vkn. This leads to a simple asymptotic formula for || Vkn|| and to a simple ‘asymptotically extremal sequence’; that is, a sequence ( un) in Lp (0, 1) with || un|| p=1 and
as n → ∞. As an application, we derive a limit theorem for large deviations, which appears to be beyond the established theory. 相似文献
3.
Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L02( G) and X is either Lp( G) for 1< p?∞ or C( G), then we show that every complete norm |·| on X that makes translations from ( X,|·|) into itself continuous is equivalent to ||·|| p or ||·|| ∞ respectively. If 1< p?∞ and every left invariant linear functional on Lp( G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on Lp( G) that makes translations from ( Lp( G),|·|) into itself continuous and that makes the map t? Lt from G into bounded is equivalent to ||·|| p. 相似文献
4.
In this paper we estimate the norm of the Moore-Penrose inverse T( a) + of a Fredholm Toeplitz operator T( a) with a matrix-valued symbol a∈ LN × N∞ defined on the complex unit circle. In particular, we show that in the ”generic case” the strict inequality || T( a) +|| > || a−1|| ∞ holds. Moreover, we discuss the asymptotic behavior of || T( tra) +|| for
. The results are illustrated by numerical experiments. 相似文献
5.
Let L
p
, 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm
|| f ||p = ( ò - pp | f |p )1 \mathord | / |
\vphantom 1 p p {\left\| f \right\|_p} = {\left( {\int\limits_{ - \pi }^\pi {{{\left| f \right|}^p}} } \right)^{{1 \mathord{\left/{\vphantom {1 p}} \right.} p}}} , and let C = L
∞ be the space of continuous 2π-periodic functions with the norm
|| f ||¥ = || f || = maxe ? \mathbbR | f(x) | {\left\| f \right\|_\infty } = \left\| f \right\| = \mathop {\max }\limits_{e \in \mathbb{R}} \left| {f(x)} \right| . Let CP be the subspace of C with a seminorm P invariant with respect to translation and such that
P(f) \leqslant M|| f || P(f) \leqslant M\left\| f \right\| for every f ∈ C. By ?k = 0¥ Ak (f) \sum\limits_{k = 0}^\infty {{A_k}} (f) denote the Fourier series of the function f, and let l = { lk }k = 0¥ \lambda = \left\{ {{\lambda_k}} \right\}_{k = 0}^\infty be a sequence of real numbers for which ?k = 0¥ lk Ak(f) \sum\limits_{k = 0}^\infty {{\lambda_k}} {A_k}(f) is the Fourier series of a certain function f
λ ∈ L
p
. The paper considers questions related to approximating the function f
λ by its Fourier sums S
n
(f
λ) on a point set and in the spaces L
p
and CP. Estimates for || fl - Sn( fl ) ||p {\left\| {{f_\lambda } - {S_n}\left( {{f_\lambda }} \right)} \right\|_p} and P(f
λ − S
n
(f
λ)) are obtained by using the structural characteristics (the best approximations and the moduli of continuity) of the functions
f and f
λ. As a rule, the essential part of deviation is estimated with the use of the structural characteristics of the function f.
Bibliography: 11 titles. 相似文献
6.
The aim of this article is to prove the following theorem.
Theorem
Let
p
be in (1,∞), ℍ
n,m
a group of Heisenberg type, ℛ the vector of the Riesz transforms on ℍ
n,m
. There exists a constant
C
p
independent of
n
and
m
such that for every
f∈ L
p
(ℍ
n,m
)
Cp-1e-0.45m||f||Lp(\mathbbHn,m) £ |||Rf|||Lp(\mathbbHn,m) £ Cpe0.45m||f||Lp(\mathbbHn,m).C_p^{-1}e^{-0.45m}\|f\|_{L^p(\mathbb{H}_{n,m})}\leq\||\mathcal{R}f|\|_{L^p(\mathbb{H}_{n,m})}\leq C_pe^{0.45m}\|f\|_{L^p(\mathbb{H}_{n,m})}. 相似文献
7.
Let X be a Banach space, (Ω,Σ, μ) a finite measure space, and L1( μ, X) the Banach space of X-valued Bochner μ-integrable functions defined on Ω endowed with its usual norm. Let us suppose that Σ 0 is a sub- σ-algebra of Σ, and let μ0 be the restriction of μ to Σ 0. Given a natural number n, let N be a monotonous norm in . It is shown that if X is reflexive then L1( μ0, X) is N-simultaneously proximinal in L1( μ, X) in the sense of Fathi et al. [Best simultaneous approximation in Lp( I, E), J. Approx. Theory 116 (2002), 369–379]. Some examples and remarks related with N-simultaneous proximinality are also given. 相似文献
8.
The well known Daugavet property for the space L
1 means that || I + K || = 1+ || K || for any weakly compact operator K : L
1 → L
1, where I is the identity operator in L
1. We generalize this theorem to the case when we consider an into isomorphism J : L
1 → L
1 instead of I and a narrow operator T. Our main result states that , where d = || J|| || J
−1||. We also give an example which shows that this estimate is exact.
Received: 21 August 2007 相似文献
9.
Let X be a (closed) subspace of Lp with 1≤ p<∞, and let A be any sectorial operator on X. We consider associated square functions on X, of the form and we show that if A admits a bounded H∞ functional calculus on X, then these square functions are equivalent to the original norm of X. Then we deduce a similar result when X= H1(ℝ N) is the usual Hardy space, for an appropriate choice of || || F. For example if N=1, the right choice is the sum for h ∈ H1(ℝ), where H denotes the Hilbert transform. 相似文献
10.
In this paper, interpolation by scaled multi-integer translates of Gaussian kernels is studied. The main result establishes
L
p
Sobolev error estimates and shows that the error is controlled by the L
p
multiplier norm of a Fourier multiplier closely related to the cardinal interpolant, and comparable to the Hilbert transform.
Consequently, its multiplier norm is bounded independent of the grid spacing when 1< p<∞, and involves a logarithmic term when p=1 or ∞. 相似文献
11.
The object of this paper is to prove the following theorem: If Y is a closed subspace of the Banach space X, then L1(μ, Y) is proximinal in L1(μ, X) if and only if Lp(μ, Y) is proximinal in Lp(μ, Y) for every p, 1 < p < ∞. As an application of this result we prove that if Y is either reflexive or Y is a separable proximinal dual space, then L1(μ, Y) is proximinal in L1(μ, X). 相似文献
12.
Let L=?Δ+|ξ| 2 be the harmonic oscillator on $\mathbb{R}^{n}Let L=−Δ+|ξ|2 be the harmonic oscillator on
\mathbbRn\mathbb{R}^{n}
, with the associated Riesz transforms R2j−1=(∂/∂ξj)L−1/2,R2j=ξjL−1/2. We give a shorter proof of a recent result of Harboure, de Rosa, Segovia, Torrea: For 1<p<∞ and a dimension free constant Cp,
||(?k=12n|Rk(f)|2)1/2||Lp(\mathbbRn,dx)\leqslant Cp||f||Lp(\mathbbRn,dx).\bigg\Vert \bigg(\sum_{k=1}^{2n}\vert R_{k}(f)\vert ^{2}\bigg)^{{1}/{2}}\bigg\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}\leqslant C_{p}\Vert f\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}. 相似文献
13.
It is shown that if A and B are positive operators on a separable complex Hilbert space, then
for every unitarily invariant norm. When specialized to the usual
operator norm ||·|| and the Schatten p-norms ||·|| p, this inequality asserts that
and
These inequalities improve upon some earlier related inequalities.
Other norm inequalities for sums of positive operators are also obtained. 相似文献
14.
For integrable functions f defined on the interval [− π, π], we denote the partial sums of the corresponding Fourier series by Sn( f) ( n=0,1,2,…). In this paper, we deal with the problem of bounding sup n|| Sn||, where ||·|| denotes an operator norm induced by a weighted L2-norm for functions f on [− π, π]. For weight functions w belonging to a class introduced by Helson and Szegö (Ann. Mat. Pura Appl. 51 (1960) 107), we present explicit upper bounds for sup n|| Sn|| in terms of w.The authors were led to the problem of deriving explicit upper bounds for sup n|| Sn||, by the need for such bounds in an analysis related to the Kreiss matrix theorem—a famous result in the fields of linear algebra and numerical analysis. Accordingly, the present paper highlights the relevance of bounds on sup n|| Sn|| to these fields. 相似文献
15.
从两个方面讨论具有最小二乘谱约束的对称斜哈密尔顿矩阵的逼近问题:(Ⅰ)研究使AX-XA的Frobenius范数最小的n阶实对称斜哈密尔顿矩阵A的集合C,其中X,A分别是特征向量和特征值矩阵, (Ⅱ)求(A)∈c使得‖C-(A)‖=min ‖C-A‖,这里‖·‖是Frobenius范数.给出了C的元素的一般表达式和(A)的显示表达式,分析了该最佳逼近矩阵A的扰动理论,并给出了数值实验. 相似文献
16.
It is proved that if positive definite matrix functions (i.e. matrix spectral densities) S
n
, n=1,2,… , are convergent in the L
1-norm, || Sn- S|| L1? 0\|S_{n}-S\|_{L_{1}}\to 0, and ò 02plogdet Sn( eiq) dq?ò 02plogdet S( eiq) dq\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S_{n}(e^{i\theta})\,d\theta\to\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S(e^{i\theta})\,d\theta, then the corresponding (canonical) spectral factors are convergent in L
2, || S+n- S+|| L2? 0\|S^{+}_{n}-S^{+}\|_{L_{2}}\to 0. The formulated logarithmic condition is easily seen to be necessary for the latter convergence to take place. 相似文献
17.
Let B1:
n×
N1→
m1, B2:
n×
N2→
m2and Q:
m2→
m1be bilinear forms which are related as follows: if μand νsatisfy B1( ξ, μ)=0 and B2( ξ, ν)=0 for some ξ≠0, then μτQν=0. Suppose p−1+ q−1=1. Coifman, Lions, Meyer and Semmes proved that, if uLp(
n) and vLq(
n), and the first order systems B1( D, u)=0, B2( D, v)=0 hold, then uτQvbelongs to the Hardy space H1(
n), provided that both (i) p= q=2, and (ii) the ranks of the linear maps Bj( ξ, ·) :
Nj→
m1are constant. We apply the theory of paracommutators to show that this result remains valid when only one of the hypotheses (i), (ii) is postulated. The removal of the constant-rank condition when p= q=2 involves the use of a deep result of Lojasiewicz from singularity theory. 相似文献
18.
Let Ω be a plane bounded region. Let U = { Uμ( P):μ( P)ε L∞(Ω), uμ ε H22, 0(Ω) and a( P, μ( P)) uμ,xx + 2 b( P, μ( P)) uμ,xy + c( P, μ( P)) uμ,vv = ƒ( P) for P ε Ω; here we are given a( P, X), b( P, X), c( P, X) ε L∞(Ω × E1), ƒ( P) ε Lp(Ω) with p > 2, and our partial differential equation is uniformly elliptic. The functions μ( P) are called profiles. We establish sufficient conditions—which when they apply are constructive—that there exist a μ 0 ε L∞(Ω) such that uμ0 ( P) uμ( P) for all P ε Ω and for each μ ε L∞(Ω). Similar results are obtained for a difference equation and convergence is proved. 相似文献
19.
Let ( E, H, μ) be an abstract Wiener space and let DV:= VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space . Given a bounded operator B on , coercive on the range , we consider the operators A:= V*BV in H and in , as well as the realisations of the operators and in Lp( E, μ) and respectively, where 1< p<∞. Our main result asserts that the following four assertions are equivalent: - (1) with for ;
- (2) admits a bounded H∞-functional calculus on ;
- (3) with for ;
- (4) admits a bounded H∞-functional calculus on .
Moreover, if these conditions are satisfied, then . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where , V= I, ). A one-sided version of (1)–(4), giving Lp-boundedness of the Riesz transform in terms of a square function estimate, is also obtained. As an application let − A generate an analytic C0-contraction semigroup on a Hilbert space H and let − L be the Lp-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an Lp-domain characterisation for the operator L.
Keywords: Divergence form elliptic operators; Abstract Wiener spaces; Riesz transforms; Domain characterisation in Lp; Kato square root problem; Ornstein–Uhlenbeck operator; Meyer inequalities; Second quantised operators; Square function estimates; H∞-functional calculus; R-boundedness; Hodge–Dirac operators; Hodge decomposition 相似文献
20.
Given a frame F = { f
j
} for a separable Hilbert space H, we introduce the linear subspace HpFH^{p}_{F} of H consisting of elements whose frame coefficient sequences belong to the ℓ
p
-space, where 1 ≤ p < 2. Our focus is on the general theory of these spaces, and we investigate different aspects of these spaces in relation
to reconstructions, p-frames, realizations and dilations. In particular we show that for closed linear subspaces of H, only finite dimensional ones can be realized as HpFH^{p}_{F}-spaces for some frame F. We also prove that with a mild decay condition on the frame F the frame expansion of any element in HFpH_{F}^{p} converges in both the Hilbert space norm and the ||·||
F, p
-norm which is induced by the ℓ
p
-norm. 相似文献
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